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The angles of depression of the top and the bottom of a 7 m tall building from the top of a tower are $45 °$ and $60 °$ respectively. What will be the height of the tower?

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7 3 + 3 2 m .

6 3 + 3 2 m .

7 3 + 3 4 m .

7 3 + 2 2 m .

answer is A.

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Detailed Solution

Given,
The angle of depression of the top and the bottom of a tall building from the top of a tower =

45 °

and

60 ° .

Height of the building=7m
We draw a figure,
Question Image

Determining the relationship between the distance between the tower and the building, and the height of the tower,
Considering

A E = h

, then

A B = h + 7

,
In triangle AED,

⇒ tan 45 ° = A E D E ⇒ tan 45 ° = h x ⇒ 1 = h x ⇒ h = x

…….(i)
In triangle ABC,

⇒ tan 60 ° = A B B C ⇒ 3 = h + 7 x ⇒ h + 7 = 3 x ⇒ h = 3 x − 7

……(ii)
Substituting the value for h in equation (ii) from equation (i),

⇒ h = 3 h − 7 ⇒ 3 h − h = 7 ⇒ h 3 − 1 = 7 ⇒ h = 7 3 − 1

The height,

⇒ A B = h + 7 ⇒ A B = 7 3 − 1 + 7 ⇒ A B = 7 + 7 (3 − 1) 3 − 1 ⇒ A B = 7 (1 + (3 − 1)) 3 − 1

On rationalizing,
Question Image

Thus the height of the tower will be

7 3 + 3 2 m .

Therefore the correct option is 1.

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